Cardinality of Sets of Positive and Negative Numbers

Is the cardinal number of the set of all positive numbers excluding zero equal to the cardinal number of the set of all negative numbers? Indeed, the two sets have the same cardinality and both are uncountably infinite.

Understanding Cardinality

In set theory, cardinality is a measure of the 'number of elements' in a set. Two sets have the same cardinality if there is a bijection (one-to-one and onto function) between them. This means that every element in one set can be paired with exactly one element in the other set, and vice versa.

Positive and Negative Numbers

The set of all positive numbers excluding zero can be represented as:

{1, 2, 3, 4, ...}

The set of all negative numbers can be represented as:

{-1, -2, -3, -4, ...}

To demonstrate that these two sets have the same cardinality, we can establish a bijection between them. One such bijection can be defined as:

1 ? -1 2 ? -2 3 ? -3 4 ? -4 ...

This mapping is a one-to-one correspondence, meaning each positive integer corresponds to a unique negative integer, and vice versa. Therefore, both sets have the same cardinality, which is countably infinite. In terms of cardinal numbers, both sets can be described with the cardinality aleph-null (??), which is the cardinality of the set of natural numbers.

Equal Cardinality with Integers and Rational Numbers

The cardinal number of the set of all positive integers is the same as that of the set of all negative integers, and both are equal to the set of all integers including zero. The addition of zero is irrelevant as it is a finite addition to the set, i.e., it only adds one more element.

To accommodate zero, we can shift each element by one. For example, for any positive integer n, we can map it to n-1. For the integer zero, we can map it to itself. This shifting effectively pairs every positive integer with a unique integer, and every integer with a unique positive integer. Hence, the set of all integers has the same cardinality as the set of all positive integers, which is ??.

Comparison with Other Sets

Interestingly, the set of positive integers (natural numbers) can be shown to be equal in cardinality to the set of odd numbers and the set of even numbers. This is because:

Every natural number can be multiplied by 2 to get a corresponding even number. Similarly, every natural number can be multiplied by 2 and then increased by 1 to get a corresponding odd number.

This means we can map the set of positive integers to the set of even numbers by multiplying each number by -2, and to the set of odd numbers by multiplying each number by 2 and adding 1. Both of these mappings are bijections and demonstrate that the cardinality of the set of positive integers is equal to that of the set of even and odd numbers.

The cardinality of the set of positive integers is also equal to the cardinality of the set of rational numbers. This is because the rational numbers can be represented as fractions, and there is a one-to-one correspondence between the set of positive integers and the set of positive rational numbers.

Conclusion

In summary, the cardinal number of the set of all positive numbers excluding zero is equal to the cardinal number of the set of all negative numbers, both of which are countably infinite and described with the cardinality ??. This property extends to the set of all integers (including zero) and also to the sets of even and odd numbers.

Further Reading

To learn more about the fascinating topic of cardinality and infinite sets, I recommend watching videos on the YouTube channel Vsauce. They provide a comprehensive and engaging explanation of these concepts in mathematics.